Asymptotic computations of tropical refined invariants in genus 0 and 1

TL;DR

The paper analyzes asymptotic refined tropical invariants for genus 0 and 1 on toric surfaces using floor diagrams. It develops a floor-diagram framework and a change of variables to x to extract asymptotics, yielding explicit generating-series expressions for the leading and first subleading terms across Hirzebruch and more general h-transverse toric surfaces, with the leading term tied to partition numbers via p(x) and the first correction controlled by the Eisenstein series E2. In genus 0 it proves AR^X_0(β)=p(x)^{χ} for these surfaces, and in genus 1 it obtains AR^X_1(β)=p(x)^{χ}(g_max-12E_2(x)) with g_max=1+½(β^2+β·K_X), plus extensions to the h-transverse setting. These results expose a regular, quasi-modular structure in the asymptotics and motivate conjectures about universal polynomials in β^2 and β·K_X, hinting at connections to GW theory with λ-classes and to relative Hilbert schemes via χ_{-y} genus. The work provides new computational tools and a clearer view of the asymptotic landscape of tropical refined invariants, suggesting broad applicability and guiding future extensions to higher genus and more general toric geometries.

Abstract

Block and Göttsche introduced a Laurent polynomial multiplicity to count tropical curves. Itenberg and Mikhalkin then showed that this multiplicity leads to invariant counts called tropical refined invariants. Recently, Brugallé and Jaramillo-Puentes studied the polynomiality properties of the coefficients of these invariants and showed that for fixed genus g, the coefficients ultimately coincide with polynomials in the homology class of the curves we look at. We call the generating series of these polynomials asymptotic refined invariant. In genus 0, the asymptotic refined invariant has been computed by the second author in the h-transverse case. In this paper, we give a new proof of the formula for the asymptotic refined invariant for g = 0 using variations on the floor diagram algorithm. This technique allows also to compute the asymptotic refined invariant for g = 1. The result exhibits surprising regularity properties related to the generating series of partition numbers and quasi-modular forms.

Figures (7)

• Figure 1:
• Figure 2: Examples of floor diagrams.
• Figure 3: The diagram $\mathcal{D}_0$.
• Figure 4:
• Figure 5: The two marked diagrams corresponding to the words from Example \ref{['expl-correspondence-word-diagram-genus-0-hirzebruch']}.

Theorems & Definitions (111)

• Theorem : \ref{['theo-ARcodeg1']}
• Theorem : \ref{['theo-AR-genus1-general']}
• Conjecture 1.1
• Remark 2.1
• Example 2.2
• Example 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Definition 2.6